Low-symmetry two-dimensional BNP 2 and C 2 SiS structures with high and anisotropic carrier mobilities
LLow-symmetry two-dimensional BNP and C SiS structureswith high and anisotropic carrier mobilities
Shixin Song, Jie Guan, ∗ and David Tom´anek † School of Physics, Southeast University, Nanjing, 211189, PRC Physics and Astronomy Department, Michigan State University, East Lansing, Michigan 48824, USA
We study the stability and electronic structure of previously unexplored two-dimensional (2D)ternary compounds BNP and C SiS. Using ab initio density functional theory, we have identifiedfour stable allotropes of each ternary compound and confirmed their stability by calculated phononspectra and molecular dynamics simulations. Whereas all BNP allotropes are semiconducting, wefind C SiS, depending on the allotrope, to be semiconducting or semimetallic. The fundamentalband gaps of the semiconducting allotropes we study range from 1 . . . × cm V − s − . Such high mobilities are quiteuncommon in semiconductors with so wide band gaps. Structural ridges in the geometry of allallotropes cause a high anisotropy in their mechanical and transport properties, promising a widerange of applications in electronics and optoelectronics. INTRODUCTION
Two-dimensional (2D) materials have intrigued re-searchers around the world since the successful mechan-ical exfoliation of graphene [1]. Even though grapheneremains unsurpassed in terms of high charge carrier mo-bility, the vanishing band gap in the pristine material pre-cludes its use from semiconducting circuitry [2, 3]. Other2D materials including transition metal dichalchogenides(TMDs) such as MoS have sizable band gaps, but arenot as useful due to their low carrier mobility [4, 5]. Phos-phorene, a monolayer of black phosphorus, combines highand anisotropic carrier mobility with a sizeable and tun-able band gap [6–8], but is unstable under ambient con-ditions [9]. In spite of significant efforts to improve theperformance of 2D materials in semiconducting devices,the progress has been moderate. There is a need to findnew 2D semiconductors with substantial band gaps andhigh carrier mobilities.Phosphorus carbide (PC), a recently proposed 2D ma-terial, has been predicted to be stable [10] and to dis-play promising electronic behavior including high carriermobility [11]. Of the stable allotropes, the semiconduct-ing α -PC phase, also called black phosphorus carbide( b -PC), has been successfully synthesized. It shows ahigh field-effect mobility µ = 1995 cm V − s − of holesat room temperature [12], good infrared response [13]and tunable anisotropic plasmonic performance [14]. Thenarrow band gap, however, limits the performance ofPC-based field-effect transistors (FETs) due to a rela-tively low ON/OFF ratio [12]. Experimental data for 2DGeP [15], another low-symmetry IV-V compound withstrongly anisotropic conductance, indicate a high field-effect ON/OFF ratio ≈ × , but a low carrier mo-bility µ = 0 .
35 cm V − s − . Theoretical explorationshave been extended to other 2D IV-V compounds, which ∗ [email protected] † [email protected] are not isoelectronic to PC. These include germaniumtriphosphide [16] GeP , tin triphosphide [17] SnP , andphosphorus hexacarbide [18] PC . However, all thesesystems have narrow band gaps similar to α -PC andshare a hexagonal honeycomb lattice structure and thusa weak transport anisotropy. 2D structures with a sub-stantial band gap, high carrier mobility and strong in-layer anisotropy are still missing.In search of such 2D materials inspired by anisotropic2D PC structures, we have applied an effective designstrategy known as “isoelectronic substitution”. This pro-cess involves substituting certain elements in the struc-ture by their neighbors in the periodic table, yet keepingthe total valence electron count unchanged. This ap-proach, which has been successfully applied in both 3Dand 2D systems, allows to change physical and chemi-cal properties of the system without drastically changingthe structure. In this way, the diamond structure of bulksilicon with a diatomic unit cell can be changed to theisoelectronic Si AlP, when one Si atom is substituted byAl and the other by P in every other unit cell, thussignificantly increasing light absorption in the visible re-gion [19]. In semimetallic 2D graphene with a diatomicbasis, substituting one C atom by B and the other by Nforms the h -BN structure, a wide-gap insulator. In a sim-ilar way, substituting every other atom in phosphoreneby Si and S atoms results in the 2D SiS structure [20, 21].The same isoelectronic substitution in 2D group V sys-tems leads to 2D group IV-VI compounds including GeS,GeSe, and SnS with a lower symmetry and a wide rangeof physical properties [22–25].In this study, we propose isoelectronic substitution in2D structures of phosphorus carbide that leads to pre-viously unknown ternary compounds BNP and C SiS.Our ab initio density functional calculations identify fourstable allotropes of each compound that share the 2D ge-ometry with PC. Calculated phonon spectra and ab initio molecular dynamics (MD) simulations confirm the stabil-ity of each of these allotropes. Due to structural ridgesin the geometry, all allotropes considered exhibit a sig- a r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Fig. 1 y xz xy xz x α β α β BNPCSiS (a) (b) (d)(c)(e) (f) (h)(g)
BNP C SiS
FIG. 1. Ball-and-stick models of relaxed monolayer structures of (a) α , (b) β , (c) α and (d) β allotropes of BNP , and (e) α , (f) β , (g) α and (h) β allotropes of C SiS in top and side view. The lattice vectors are indicated by red arrows. nificant anisotropy in the elastic response. Electronicstructure calculations indicate that α and β phases ofC SiS, as well as all four BNP allotropes, are semicon-ductors with a wide range of band gap values. Carriermobility calculations show that most of the semiconduct-ing allotropes exhibit high and strongly anisotropic car-rier mobilities. The remaining α -C SiS and β -C SiSallotropes were found to be semimetallic, displaying anelliptically distorted Dirac cone in their band structurecaused by their structural anisotropy.
Fig. S7Strain (%) Δ E ( e V / a t o m ) (a) BNP α β α β xy (b) C SiS α β α β xy Strain (%) -10 -8 -6 -4 -2 0 2 4 6 8 10-10 -8 -6 -4 -2 0 2 4 6 8 10
FIG. 2. Effect of uniaxial in-layer strain on the relative bind-ing energy ∆ E in monolayers of (a) BNP and (b) C SiS.Results for different allotropes are distinguished by color andsymbols. Results for strain along the x -direction are shownby solid lines and for strain along the y -direction by dashedlines. COMPUTATIONAL TECHNIQUES
We have used ab initio density functional theory(DFT), as implemented in the
VASP code [26–28],throughout the study. We applied periodic boundaryconditions, with 2D structures separated by a vacuumregion in excess of 20 ˚A. The reciprocal space was sam-pled by a fine grid [29] of 8 × × k -points in the Brillouinzone of 8-atom unit cells or its equivalent in supercells.We used projector-augmented-wave (PAW) pseudopoten-tials [30] and the Perdew-Burke-Ernzerhof (PBE) [31]exchange-correlation functionals. Selective band struc-ture calculations were performed using the hybrid HSE06functional [32, 33] with the default mixing parameter α = 0 .
25. We utilized the DFT-D2 [34] method to rep-resent van der Waals (vdW) corrections to the total en-ergy. We used a cutoff energy of 500 eV for the plane-wave basis set and considered the electronic structure tobe converged once the total energy difference betweensubsequent electronic structure iterations would not ex-ceed 10 − eV. All geometries were optimized using theconjugate-gradient method [35], until none of the resid-ual Hellmann-Feynman forces exceeded 10 − eV/˚A. Thephonon calculations were carried out using the densityfunctional perturbation theory [36–38], as implementedin the PHONOPY code [39]. The system dynamics wasstudied using canonical ab initio
MD simulations with3 fs time steps. Our results are for supercells containingmore than 100 atoms, which were kept at temperaturesof 300 K, 500 K or 1000 K for periods exceeding 15 ps.
TABLE I. Calculated equilibrium properties of BNP and C SiS 2D allotropes.BNP C SiS α β α β α β α β | (cid:126)a | (˚A) a | (cid:126)a | (˚A) a d P − P (˚A) b h h i i d P − N (˚A) b d P − B (˚A) b d B − N (˚A) b d Si − S (˚A) b h h i i d C − S (˚A) b d C − Si (˚A) b d C − C (˚A) b E coh (eV/atom) c E form (eV/atom) d -0.01 -0.01 -0.25 -0.24 0.68 0.68 0.39 0.39 c (N/m) e c (N/m) e E g − PBE (eV) f E g − HSE (eV) g a | (cid:126)a | and | (cid:126)a | are the in-plane lattice constants defined in Fig. 1. b d is the equilibrium bond length between the respective species. c E coh is the cohesive energy per average atom with respect to isolated atoms. d E form is the formation energy per average atom with respect to elemental structures. e c ( c ) are the 2D elastic constants along the x ( y ) direction. f E g -PBE is the band gap value at the DFT-PBE level. g E g -HSE06 is the band gap value at the DFT-HSE06 level. h Length of bonds within the structural plane. i Length of bonds out of the structural plane.
RESULTS AND DISCUSSIONStructure of BNP and C SiS 2D allotropes
As introduced above, the 2D compounds BNP andC SiS are isoelectronic to previously reported PC mono-layer structures [10]. We have identified four 2D al-lotropes, called α , β , α and β , for each of the sys-tems. The most stable structures of each allotrope areshown in Fig. 1. All atoms are 3-fold coordinated, caus-ing a coexistence of sp and sp bonding and leading tostructural ridges in the geometry of all allotropes. Whenviewed from the side, the α allotropes have an armchairprofile, whereas the β allotropes have a zigzag profile.With the exception of β -C SiS with 8 atoms in the unitcell, all allotropes have rectangular unit cells containing16 atoms. Additional metastable structures with differ-ent atomic arrangements in the unit cell are discussed inthe Appendix.As seen in Fig. 1, the optimized α and β allotropes ofBNP structures consist of isolated P-P and B-N dimersforming a 2D hexagonal structure. Since the orientationof the B-N dimers alternates in the plane of the system,there is no net dipole moment in the system. P atomsprefer the sp configuration with a lone electron pair,whereas B and N atoms prefer the sp configuration in all BNP allotropes. The α and β allotropes representtwo equivalent ways to achieve optimum configurationwith the same topology, in analogy to black and bluephosphorene. [40]The α and β allotropes of BNP contain alternatingchains of P and BN along the y -direction. The mismatchbetween the equilibrium P-P and B-N bond lengths leadsto the formation of pentagon-heptagon pairs instead ofhexagons.The structure of all 2D allotropes of C SiS closelyresembles that of BNP due to the coexistence of sp -bonded C atoms and sp -bonded Si and S atoms. Simi-lar to BNP , the α and β allotropes of C SiS containisolated Si-S and C-C dimers. Also in this case, there aretwo distinguishable, topologically equivalent geometries.Even though the polar Si-S bonds are out of plane inthis compound, their alternating orientation eliminates anet dipole moment. The α and β allotropes of C SiS,analogous to those of BNP , contain alternating C andSiS chains along the y -direction. Bond length mismatchleads to a preferential formation of pentagon-heptagonpairs in the layer.A summary of the structural characteristics and thecohesive energy of all these allotropes is presented in Ta-ble I. In the BNP system, the B-N dimers in α and β allotropes are connected by typical double bonds with FIG. 3. Electronic structure of a BNP monolayer. Shown are the calculated band structure E ( k ) and the density of states(DOS) including its projection on individual atoms (left panels), the partial charge density distributions at the valence bandmaximum (VBM) and the conduction band minimum (CBM) (right panels) of the (a) α , (b) α , (c) β , and (d) β allotropes.All results are based on the DFT-PBE functional with the exception of DFT-HSE06 data, which are shown by the red dashedlines in the band structure plots. The blue arrows indicate the fundamental band gap. Contributions of different atoms aredistinguished by color in the PDOS plots. Isosurface plots of the partial charge density are presented at the isosurface value of8 . × − e/Bohr for all allotropes. d BN ≈ .
40 ˚A, close to the 1 .
403 ˚A value in amino bo-rane [41]. The bond length of the B-N chains in α and β allotropes is slightly longer, at about 1 .
46 ˚A, close tothe 1 .
45 ˚A value in 2D h -BN [42]. These results are con-sistent with the sp configuration of B and N atoms in2D BNP .We find similar structural characteristics in the C SiSsystem. As expected for sp -bonded C atoms, the C-Cbond lengths are comparable to the double bonds in ethy-lene and the conjugated bonds in benzene or graphene.The length of all the other single bonds appears ratherinsensitive to the structure.It is also noteworthy that P-N bonds are slightlyshorter than P-B bonds in BNP due to the smalleratomic radius of the more electronegative element. Thesame is true, for the same reason, with the C-S bonds thatare slightly shorter than C-Si bonds in C SiS allotropes.For the sake of easy comparison, we define the cohesiveenergy E coh of an “average” atom by dividing the totalatomization energy by the total number of atoms. Ourresults for E coh are listed in Table I. We found that forboth BNP and C SiS, α and β allotropes with the sameindex are almost equally stable. Our results also indicatethat for both BNP and C SiS, α and β allotropes areenergetically more stable than α and β allotropes by∆ E coh ≈ . − . SiS structuresto be generally more stable than BNP structures. The listed values of E coh ≈ ab initio MD simulations. Details of all thesecalculations are presented in the Appendix.We also calculated the formation energy per ‘averageatom’ E form to investigate the relative stability of theternary allotropes comparing with respect to their ele-mental components. We defined E form as E form ( X l Y m Z n ) = E ( X l Y m Z n ) − l · E ( X ) + m · E ( Y ) + n · E ( Z ) l + m + n , (1)where E ( X l Y m Z n ), E ( X ), E ( Y ), E ( Z ) are the respec-tive total energies per average atom of the X l Y m Z n com-pound and the elemental structures of X, Y and Z. In theparticular case of 2D BNP and C SiS structures, the el-emental structures we consider are bulk boron, gas of N molecules, phosphorene, graphene, bulk silicon and bulksulfur. The results of E form for all the allotropes arelisted in Table I. E form > E form < E form in Table I indicate that allBNP allotropes are stable and will not decompose be-low ≈ FIG. 4. Contour plots of the energy dispersion E ( k ) of thetop valence band (left panels) and the lowest conduction band(right panels) in (a) α -BNP and (b) β -BNP . The Fermilevel is set as the reference energy of zero. The values of E form for C SiS allotropes considered hereare positive, but rather small. The values E form =0 .
68 eV/atom for the α and β phases and E form =0 .
39 eV/atom for the α and β phases are an orderof magnitude smaller than cohesive energies and com-parable to E form = 0 .
54 eV/atom found in 2D-PC [10].Structural changes would require overcoming a signifi-cant activation barrier associated with breaking bondsand protect C SiS allotropes from structural changes atroom temperature. As mentioned above, interatomicbonds may break and structural changes may becomepossible at extremely high temperatures of ≈ Anisotropic in-plane stiffness of 2D BNP and C SiSallotropes
To investigate the elastic response of the 2D ternarystructures, we subjected all allotropes in this study to in-layer uniaxial strain along the x - and y -direction and dis-play differences ∆ E coh of the average binding energy withrespect to the most stable allotrope in Fig. 2. The pres-ence of structural ridges renders all allotropes much softeralong the x -direction normal to the ridge direction. The α allotropes of both BNP and C SiS are particularlysoft along the x -direction, with ∆ E (cid:46)
18 meV/atom whensubjected to ≤
10% compressive or tensile strain. The2D elastic moduli [43] c ii along the x - and y -direction,obtained using these calculations, are summarized in Ta-ble I. For the semiconducting allotropes, these values playan important role in carrier mobilities. Electronic structure of BNP
2D allotropes
Results of our DFT calculations for the electronicstructure of BNP monolayers are presented in Fig. 3.Our DFT-PBE results show all the four allotropes to besemiconductors. The fundamental band gap is direct in α -BNP and indirect in all the other BNP allotropes.Band gap values E g based on DFT-PBE Kohn-Sham en-ergies, ranging from 0 .
52 eV to 1 .
39 eV, are listed inTable I. We should note here that the interpretation ofKohn-Sham eigenvalues as self-energies is strictly incor-rect and that DFT-based band gaps are typically under-estimated. We find this to be the case when compar-ing band gap values based on DFT-PBE and the hybridDFT-HSE06 functional in Table I. As seen when com-paring band structure results based on DFT-PBE, givenby the solid lines in Fig. 3, and DFT-HSE06, shown bythe red dashed lines, the main difference is the opening ofthe fundamental band gap in DFT-HSE06, whereas theband dispersion is unaffected. We also present the totalelectronic density of states (DOS) and its projection ontothe different species next to the band structure plots.Contour plots of the projected DOS (PDOS) of allBNP allotropes at the valence band maximum (VBM)and conduction band minimum (CBM) are shown inFig. 3, with the contributions to VBM and CBM dis-tinguished by color. We find the character of the VBMto be dominated by the lone-pair states of the sp hy-bridized P atoms in all BNP allotropes. The characterof the CBM is very different. The dominant contributionin α and β allotropes of BNP comes from out-of-plane p orbitals of sp hybridized B and N atoms. In the α and β allotropes, p orbitals of all the atoms contributeto the CBM character.We also observe an unusually flat band in the bandstructure of α and β allotropes of BNP at the topof the valence region, along the Γ − X line. To furtherexplore the origin of this flat band, we show 2D contourplots of the dispersion within the top valence and bottomconduction bands in Fig. 4.In both allotropes, the CBM energy minimum occursat the Γ point and the band dispersion around is al-most isotropic and free-electron like. In contrast, theVBM is displaced from the Γ point in the direction to-wards X for both allotropes. Whereas the band disper-sion is almost flat along the x -direction, it is significantlyhigher along y -direction near the VBM, indicating a sig-nificant anisotropy in the carrier effective mass. As seenin Fig. 3(b) and (d), this is caused at the VBM by alarger separation and lower hybridization between the Plone pair states along the x -direction than along the y -direction. Electronic structure of C SiS 2D allotropes
Results of our DFT calculations for the electronicstructure of C SiS monolayers are presented in Fig. 5.
FIG. 5. Electronic structure of a C SiS monolayer. The calculated band structure E ( k ) and the density of states (DOS)including its projection on individual atoms are shown in the left panels. The partial charge density distributions at the valenceband maximum (VBM) and the conduction band minimum (CBM) for the semiconducting allotropes and for the frontier statesin the semimetallic allotropes are shown in the right panels. Results are presented for the (a) α , (b) α , (c) β , and (d) β allotropes. All results are based on the DFT-PBE functional with the exception of DFT-HSE06 data, which are shown bythe red dashed lines in the band structure plots. The blue arrows indicate the fundamental band gap in the semiconductingallotropes. Contributions of different atoms are distinguished by color in the PDOS plots. Isosurface plots of the partial chargedensity are presented at the isosurface value of 8 . × − e/Bohr for all allotropes. Our findings indicate that only the α and the β al-lotropes are semiconducting, whereas the α and β al-lotropes are semimetallic. The band gap is indirect in α -C SiS and direct in β -C SiS. Numerical band gapvalues based on DFT-PBE and DFT-HSE06 are listed inTable I and fall in the range of values found in BNP .According to Fig. 5, the character of VBM and CBMstates in the semiconducting allotropes α and β is verysimilar. In analogy to results for BNP , the VBM ofC SiS is dominated by lone-pair orbitals of Si atoms andthe CBM contains mostly out-of-local-plane p orbitals ofC atoms.The α and β allotropes of C SiS are found to besemimetallic by both DFT-PBE and DFT-HSE06. Dirac“cones” are seen in the band structure in Fig. 5(b) and(d), with the Dirac point at E F located between Γ and Y in the Brillouin zone. The frontier states at E F aredominated by out-of-plane p orbitals on C atoms.The Dirac “cones” of α - and β -C SiS are visualizedin more detail in Fig. 6. The band dispersion in bothallotropes is linear around E F , but anisotropic. As seenin Fig. 6(c) and (d), the cross-section of the “cone” at E < E F or E > E F is not a circle, but rather an el-lipse elongated in the k x -direction. The higher steep- ness of the band dispersion along the y -direction nearthe Dirac point is a consequence of a stronger interactionbetween C2 p orbitals within zigzag chains of carbons,aligned with the y -direction. These C chains in α -C SiSand β -C SiS are clearly visible in Figs. 1(g) and (h).Otherwise, these “cones” are very similar in Dirac pointlocation, band dispersion and its anisotropy in both al-lotropes.
Carrier mobilities in semiconducting BNP andC SiS 2D allotropes
Anisotropy in semiconducting BNP and C SiS 2D al-lotropes is not only limited to the geometry, the elasticresponse and the electronic structure, but is also presentin the carrier mobility. In absence of defects and externalscattering centers, the mobility of carriers in 2D semicon-ductors is limited by acoustic phonons. We calculated thecarrier mobilities of BNP and C SiS 2D systems alongthe x - and y -direction using the deformation potentialtheory expression [11, 44, 45] µ i = e (cid:126) c ii k B T m ∗ i m d E i . (2) TABLE II. Calculated carrier mobilities and related quantities in semiconducting BNP and C SiS allotropes. m ∗ xa m ∗ ya E xb E yb µ xc µ yc ( m ) (eV) (10 cm V − s − ) α -BNP e h β -BNP e h α -BNP e h β -BNP e h α -C SiS e h β -C SiS e h e and holes by h . a m ∗ x ( m ∗ y ) are the carrier effective masses along the x ( y ) direction. b E x ( E y ) are the deformation potentials along the x ( y ) direction. c µ x ( µ y ) are the carrier mobilities along the x ( y ) direction at 300 K. Here, i represents the Cartesian direction, with i = 1standing for x and i = 2 for y , and e is the carriercharge. c ii is the 2D elastic modulus [43] along thedirection i , obtained from the strain energy curve inFig. 2. T is the temperature, m ∗ i is the effective massalong the i -direction, and m d is the average effectivemass given by m d = ( m ∗ x m ∗ y ) / . The deformation po-tential E i along the i -direction is determined at the va- FIG. 6. Band dispersion in the semimetallic allotropes α -C SiS (left) and β -C SiS (right) near the Fermi level. E ( k )results are presented as 3D plots in (a),(b) and 2D contourplots in (c),(d). The energy with respect to E F = 0 is repre-sented by color. lence band maximum (VBM) for holes and the conduc-tion band minimum (CBM) for electrons. It is definedby E i = ∆ V / (∆ a i /a i ), where ∆ V is the energy shift ofthe band edge with respect to the vacuum level under asmall change ∆ a i of the lattice constant a i . Our resultsfor room-temperature carrier mobilities in all semicon-ducting 2D allotropes are summarized in Table II.We found that most allotropes in our study exhibitvery high and remarkably anisotropic carrier mobilities.The highest carrier mobility value we found is µ y =1 . × cm V − s − for electrons in α -BNP along the y -direction, over two orders of magnitude higher than thevalue µ x = 0 . × cm V − s − along the x -direction.An important contributing factor to the mobility is the2D elastic modulus [43]. The superior electron mobilityalong its y -direction is a result of increased stiffness alongthis direction. According to Table I, c is indeed muchlarger than c for most allotropes we study.Unlike for electrons, hole mobility is highest along the x -direction in β , α and β allotropes of BNP . Thechange of preferential transport direction is caused bysubstantial deformation potential anisotropy E x < E y at the VBM according to Table II. Carrier mobilities insemiconducting 2D allotropes of C SiS are comparableto those in BNP .Carrier mobilities µ > cm V − s − in α -BNP with E g > . µ (cid:38) cm V − s − in α and β allotropes of BNP with E g > µ = 0 . × cm V − s − , whichhas been reported for phosphorene [6]. These results arevery promising for the realization of 2D semiconductingdevices with high ON/OFF ratios and on-state currents. SUMMARY AND CONCLUSIONS
In summary, we have introduced previously unexplored2D ternary compounds BNP and C SiS as isoelectroniccounterparts of 2D PC structures. Using ab initio den-sity functional theory, we have identified four stable al-lotropes of each compound and confirmed their stabilityby calculated phonon spectra and molecular dynamicssimulations. All allotropes display structural and elas-tic anisotropy due to structural ridges in their geometry,which are caused by coexisting sp and sp hybridiza-tion. Whereas all BNP allotropes are semiconducting,we find only two allotropes of C SiS to be semiconduct-ing. The other two allotropes of C SiS are semimetallicand show anisotropic Dirac cones at E F . The fundamen-tal band gaps of the semiconducting allotropes we studyrange from 0 . . . . . × cm V − s − .Such high mobilities, with two orders of magnitude inanisotropy ratio, are desirable, but quite uncommon insemiconductors with so wide band gaps. Combination ofwide band gaps with high and anisotropic carrier mobili-ties offer great promise for applications of 2D BNP andC SiS structures in electronics and optoelectronics.
ACKNOWLEDGMENTS
This study was supported by the National NaturalScience Foundation of China (NSFC) under Grant No.61704110, the Fundamental Research Fund for the Cen-tral Universities, the Shuangchuang Doctoral Programof the Jiangsu Province, and by the Zhongying YoungScholar Program of Southeast University. D.T. acknowl-edges financial support by the NSF/AFOSR EFRI 2-DARE grant number EFMA-1433459. We thank the BigData Computing Center of Southeast University for pro-viding facility support for performing calculations pre-sented in this manuscript.
APPENDIXMetastable structures of 2D BNP and C SiS
As seen in Fig. 1 of the main text, α and β allotropesof BNP contain four polar B-N dimers aligned with the x -axis in each unit cell. Each of these dimer bonds maybe rotated by 180 ◦ . This gives rise to four metastablestructures for α -BNP and another four structures for β -BNP , shown in Fig. A1. The most stable structuresamong these, with the cohesive energy value highlightedin red, have been considered in the main text. Fig. S1 α -BNP -1 α -BNP -2 α -BNP -3 α -BNP -4 E coh (eV/atom) E coh (eV/atom) β -BNP -1 β -BNP -2 β -BNP -3 β -BNP -4 y x BNP y x
FIG. A1. Ball-and-stick model of metastable structures re-lated to the α and β allotropes of BNP , shown in top view.Cohesive energies E coh are presented below each structure,with the value for the most stable configuration highlightedin red. Fig. S2 α -BNP -1 α -BNP -2 E coh (eV/atom) E coh (eV/atom) β -BNP -1 β -BNP -2 y x BNP y x
FIG. A2. Ball-and-stick model of metastable structures re-lated to the α and β allotropes of C SiS, shown in top view.Cohesive energies E coh are presented below each structure,with the value for the most stable configuration highlightedin red. Fig. S3 α -C SiS-1 α -C SiS-2 α -C SiS-3β -C SiS-1 β -C SiS-2 β -C SiS-3 E coh (eV/atom) E coh (eV/atom) y x CSiS y x
FIG. A3. Ball-and-stick model of metastable structures re-lated to the α and β allotropes of BNP , shown in top view.Cohesive energies E coh are presented below each structure,with the value for the most stable configuration highlightedin red. Γ ΓX S Y Γ ΓX S Y Γ ΓX S Y Γ ΓX S YΓ ΓX S Y Γ ΓX S Y Γ ΓX S Y Γ ΓX S Y α β α β BNP C SiS F r e qu e n cy ( c m - ) F r e qu e n cy ( c m - ) (a) (b) (c) (d)(e) (f) (g) (h) Fig. S4
FIG. A4. Phonon spectra for monolayers of (a) α -BNP , (b) β -BNP , (c) α -BNP , (d) β -BNP , (e) α -C SiS, (f) β -C SiS,(g) α -C SiS, and (h) β -C SiS.
Structure at 15 psStructure at 15 ps0 3 6 9 12 15
Time (ps) α -BNP E ( e V / s up e r ce ll ) E ( e V / s up e r ce ll ) -950-940-930-920-910-900-980-970-960-950-940-930 0 3 6 9 12 15 Time (ps) -980-970-960-950-940-930-980-970-960-950-940-930 α -BNP β -BNP β -BNP (a) (b)(c) (d) Fig. S5
FIG. A5. Fluctuations of the total potential energy (left panels) and structural snap shots after 15 ps (right panels) for (a) α -BNP , (b) α -BNP , (c) β -BNP , and (d) β -BNP monolayers during canonical MD simulations at 300 K, 500 K and1000 K. Quite different is the arrangement of B and N atoms inzigzag chains in the α and β allotropes of BNP shownin Fig. 1 of the main text. Also in this case, each bondmay be rotated by 180 ◦ . This gives rise to two metastablestructures for α -BNP and another two structures for β -BNP , shown in Fig. A2. Only the most stable amongthese allotropes have been considered in the main text.Next we turn to the α and β allotropes of C SiS,displayed in Fig. 1(e) and (f) of the main text. Unlikein BNP , we can find only three inequivalent metastable0 E ( e V / s up e r ce ll ) -930-920-910-900-890-880 Time (ps) E ( e V / s up e r ce ll ) -930-920-910-900-890-880 Structure at 15 ps0 3 6 9 12 15 Time (ps) α -C SiS α -C SiSβ -C SiS β -C SiS (a) (b)(c) (d)
Fig. S6
FIG. A6. Fluctuations of the total potential energy (left panels) and structural snap shots after 15 ps (right panels) for (a) α -C SiS, (b) α -C SiS, (c) β -C SiS, and (d) β -C SiS monolayers during canonical MD simulations at 300 K, 500 K and1000 K.FIG. A7. Carrier mobility versus band gap for different 2D materials. Results for semiconducting allotropes predicted in thisstudy, shown by the red symbols, are compared to previously reported values for selected 2D materials, shown by the blacksymbols. All results have been obtained at the PBE level of DFT. structures for these systems, displayed in Fig. A3. Forthe α and β allotropes of C SiS, optimization leadsto only one stable structure. As for the other systemsdiscussed above, we only consider only the most stableallotropes in the main text.
Phonon spectra of 2D BNP and C SiS allotropes
A real confirmation of structural stability, more impor-tant than a high cohesive energy, comes from the phononspectra. Structures can be considered stable if no imag-inary frequencies can be found in the phonon spectra.1The calculated phonon spectra of all BNP and C SiS al-lotropes discussed in this study are displayed in Fig. A4.We should note at this point that phonon calculationsfor 2D structures with the uniquely soft flexural ZA modeare very demanding on precision [46]. We found ourphonon spectra to be essentially free of imaginary fre-quencies associated with decay modes. The highest imag-inary frequency value ω (cid:46) i cm − in our spectra belongsto the ZA mode near the Γ point in the Brillouin zoneand is an artifact related to numerical precision [46]. Thermodynamic stability of 2D BNP and C SiSallotropes
Whereas phonon spectra tell about structural stabil-ity in the harmonic regime, they cannot determine if astructure will or will not fall apart at a given finite tem-perature. To study the thermodynamic stability of theternary structures in this study, we performed a set ofcanonical ab initio molecular dynamics (MD) simulationsat 300 K, 500 K and 1000 K and present our results inFig. A5 for 2D BNP and Fig. A6 for C SiS structures.For 15 ps long MD runs, we plotted both the fluctua-tions of the total potential energy and snapshots of thestructures after 15 ps.Our results in Fig. A5 indicate that all four allotropesof BNP maintained their geometries up 1000 K, indicat-ing a high thermodynamic stability. The correspondingresults for C SiS in Fig. A6 indicate all allotropes to be stable at 300 K and 500 K. Further increase of tempera-ture to 1000 K causes a dramatic degradation of the α and β allotropes of C SiS after 3 ps. The α and β allotropes appear more stable at 1000 K, but their sig-nificant distortion indicates onset of degradation. Thethermodynamic stability of all the BNP and C SiS al-lotropes is either comparable to or superior to other sim-ilar 2D structures including GeP [16], which was shownto be stable at T (cid:46)
500 K, as well as PC [11] and binary V-V compounds [47], which were shown to be stable onlyup to ≈
300 K. Our MD results show good consistencywith the conclusions obtained from the formation energycalculations discussed in the main text.
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